Index: src/wp-admin/includes/class-wp-site-health-auto-updates.php
===================================================================
--- src/wp-admin/includes/class-wp-site-health-auto-updates.php	(revision a0ca5afd8977b5a3857084d9cb1bd345166e2f21)
+++ src/wp-admin/includes/class-wp-site-health-auto-updates.php	(date 1554489631238)
@@ -106,6 +106,17 @@
 
 		$test = wp_remote_get( $url, compact( 'cookies', 'headers', 'timeout' ) );
 
+		if ( is_wp_error( $test ) ) {
+			return array(
+				'description' => sprintf(
+					/* translators: %s: Name of the filter used. */
+					__( 'A failure prevents us from confirming that the %s filter is available.' ),
+					'<code>wp_version_check()</code>'
+				),
+				'severity'    => 'warning',
+			);
+		}
+
 		$response = wp_remote_retrieve_body( $test );
 
 		if ( 'yes' !== $response ) {